1. ## Roulette question?

Hi everyone, brand new here and wondering if you could help settle a discussion myself and a friend had recently regarding probablity while playing roullette!

We had both been playing the game in a local casino for a while, and I had covered my chosen number of 17 for most of the afternoon that we were in there, it failed to come up at all! My friend then stated, 'it has to come eventually' and I disagreed by stating that the wheel could spin for the rest of time, for infinity and it doesnt 'have to come at all'. To which my friend stated, 'but if that happened it would definatly come' my reply being 'yes, it would probably come billions of times in infinity, but mathamatically it doesnt have to come at all!' Could you tell me which one of us is correct?

2. Originally Posted by djceney
Hi everyone, brand new here and wondering if you could help settle a discussion myself and a friend had recently regarding probablity while playing roullette!

We had both been playing the game in a local casino for a while, and I had covered my chosen number of 17 for most of the afternoon that we were in there, it failed to come up at all! My friend then stated, 'it has to come eventually' and I disagreed by stating that the wheel could spin for the rest of time, for infinity and it doesnt 'have to come at all'. To which my friend stated, 'but if that happened it would definatly come' my reply being 'yes, it would probably come billions of times in infinity, but mathamatically it doesnt have to come at all!' Could you tell me which one of us is correct?

Unless there was something which completely prevented it from landing on number 17, it will definitely come up if the wheel is spun an infinite number of times.

The infinite sum of the geometric distribution supports this. It will definitely happen, provided it DOES have a chance of happening. (Which it does )

3. Hello, djceney!

Welcome aboard!

Your friend will be disappointed, even angry, but you are correct.

I'll assume the wheel has 1-to-36, plus 0 and 00.

At every play, the probability of getting 17 is $\frac{1}{38}$
. . regardless of what has happened in the past.

You friend is tempted by the phrases "the law of averages" or "in the long run",
. . a common and understandable misconception.

Think of flipping a coin.
It is possible that you get ten Heads in a row.
. . (The probability of 10 Heads is quite small, but imagine that it has happened.)
What is the probability of getting a Tail on the next flip?

You might think that the chance of getting a Tail is in your favor now.
. . But on any flip, the probability of a Tail is still ½.

You see, the coin doesn't know (or care) what happened in the previous ten flips.
. . It is always: . $Pr(\text{Head}) = \frac{1}{2},\;\;Pr(\text{Tail}) = \frac{1}{2}$

So at the roulette table, your probability of winning is $\frac{1}{38}$ at every spin,

4. Originally Posted by janvdl
Unless there was something which completely prevented it from landing on number 17, it will definitely come up if the wheel is spun an infinite number of times.

The infinite sum of the geometric distribution supports this. It will definitely happen, provided it DOES have a chance of happening. (Which it does )
Hiya, thanks for the reply, however, you state that 'It will definitely happen, provided it DOES have a chance of happening' yes it does have a chance of happening, there is always going to be a 1/37 chance of it coming the next spin. On the same basis there is a 37/1 chance it wont come on the next spin ie : there are two outcomes available 'it lands' or 'it misses' each spin starting fresh with no memory of the previous spinned number, just as if you flip a coin 10 times and it comes up heads its no more likely to be tails on the next flip than heads, so there is a chance that 17 will never come. So just as there is a chance it might come there's also a chance it wont!

5. Originally Posted by djceney
Hiya, thanks for the reply, however, you state that 'It will definitely happen, provided it DOES have a chance of happening' yes it does have a chance of happening, there is always going to be a 1/37 chance of it coming the next spin. On the same basis there is a 37/1 chance it wont come on the next spin ie : there are two outcomes available 'it lands' or 'it misses' each spin starting fresh with no memory of the previous spinned number, just as if you flip a coin 10 times and it comes up heads its no more likely to be tails on the next flip than heads, so there is a chance that 17 will never come. So just as there is a chance it might come there's also a chance it wont!
The chances it does NOT come up is $\frac{36}{37}$ not $\frac{37}{1}$.

If it has a probability, the event WILL happen if the wheel is spun an infinite number of times.

If the wheel is spun a finite amount of times, there is always the outcome that the event will not happen at all.

$\sum_{k=1}^{\infty} (1-p)^{k-1}p = 1$

Where p is the probability. ( Given by you as 1/37 )

k is the number of times you spin the wheel until you reach a succes. The formula says if you spun it an infinite number of times, the probability = 1 = 100%. Meaning it will come up.

6. I now actually agree with you, if the argument was ' you could spin it 100 billion times and 17 doesnt has to show' then that would be correct?

7. Originally Posted by djceney
I now actually agree with you, if the argument was ' you could spin it 100 billion times and 17 doesnt has to show' then that would be correct?
You must understand while that is highly unlikely, it is possible yes.

(I edited my post to provide you with the mathematical foundation for my argument.)

8. Originally Posted by janvdl
You must understand while that is highly unlikely, it is possible yes.

(I edited my post to provide you with the mathematical foundation for my argument.)
Highly unlikely, and to be truthfull 17 would almost certainly come up a few times in the next 100 or so spins, but still mathamatically possible that it wont!

Thanks for the replies !!

9. Originally Posted by djceney
Hi everyone, brand new here and wondering if you could help settle a discussion myself and a friend had recently regarding probablity while playing roullette!

We had both been playing the game in a local casino for a while, and I had covered my chosen number of 17 for most of the afternoon that we were in there, it failed to come up at all! My friend then stated, 'it has to come eventually' and I disagreed by stating that the wheel could spin for the rest of time, for infinity and it doesnt 'have to come at all'. To which my friend stated, 'but if that happened it would definatly come' my reply being 'yes, it would probably come billions of times in infinity, but mathamatically it doesnt have to come at all!' Could you tell me which one of us is correct?

It might surprise you to know that if the wheel was spun an infinite number of times, it is mathematically certain that a sequence of N 17's in a row will occur, where N is a finite arbitrary number (1 one trillion, for example). Furthermore, this sequence will occur infinitely often with certainty.

10. Originally Posted by janvdl
The chances it does NOT come up is $\frac{36}{37}$ not $\frac{37}{1}$.

If it has a probability, the event WILL happen if the wheel is spun an infinite number of times.

If the wheel is spun a finite amount of times, there is always the outcome that the event will not happen at all.

$\sum_{k=1}^{\infty} (1-p)^{k-1}p = 1$

Where p is the probability. ( Given by you as 1/37 )

k is the number of times you spin the wheel until you reach a succes. The formula says if you spun it an infinite number of times, the probability = 1 = 100%. Meaning it will come up.
A probability of 1 does not mean that it will happen just as a probability of 0 does not mean that it cannot happen.

RonL